Question

F(x)={

3x−15
15−3x
​

if 0≤x<5
if 5≤x≤10
​
Evaluate the definite integral by interpreting it in terms of signed area. ∫
0
10
​
f(x)dx=

Answer 1

The definite integral ∫_{0}¹⁰ f(x)dx is equal to 60.The signed area under the curve of the function f(x) = {

3x − 15 (0 ≤ x < 5)

15 − 3x (5 ≤ x ≤ 10)

} from 0 to 10 can be visualized as the combination of two regions. The first region, corresponding to the interval [0,5), represents the area above the x-axis minus the area below the x-axis, resulting in a net positive signed area.

Step-by-step explanation:

The given function f(x) is defined differently for two intervals: [0,5) and [5,10]. To evaluate the definite integral ∫_{0}^{10} f(x)dx, we need to split the integral into two parts corresponding to these intervals and then calculate each part separately.

For the interval [0,5), the function is 3x - 15. The definite integral for this interval is ∫_{0}^{5} (3x - 15)dx. To find the antiderivative, we integrate each term separately:
`∫_(0)^(5) 3x dx - ∫_(0)^(5)`15 dx. Evaluating these integrals, we get
`[(3/2)x^2]_(0)^(5) - [15x]_(0)^(5),` which simplifies to (3/2) * 5^2 - 15 * 5 - (0 - 0) = 37.5.

For the interval [5,10], the function is 15 - 3x. The definite integral for this interval is ∫_{5}^{10} (15 - 3x)dx. Integrating each term separately, we get
`[15x - (3/2)x^2]_(5)^(10),` which simplifies to 150 - (75/2) - (75 - 37.5) = 22.5.

Adding these two results together, we get the final answer: 37.5 + 22.5 = 60. Therefore, ∫_{0}
`^(10)`f(x)dx = 60, representing the signed area under the curve.